**π‘ Problem Formulation:** Suppose you need to determine how many 2×1 rectangles can fit within a larger n x m rectangle. This problem is common in computational geometry and has practical applications in areas like tiling, resource allocation, and game development. For example, if you have a rectangle of size 6×4, you would want to know how many 2×1 rectangles can be placed inside it without overlapping and within the bounds.

## Method 1: Brute Force Iteration

This method involves iterating over the entire area of the larger rectangle and counting the possibilities of placing the smaller rectangle both horizontally and vertically, without counting the overlapping positions. It’s simple to implement and understand.

Here’s an example:

def count_rectangles(n, m): horizontal_fits = (n // 2) * m vertical_fits = (m // 2) * n return horizontal_fits + vertical_fits print(count_rectangles(6, 4))

**Output:** 18

This code snippet defines a function `count_rectangles`

which calculates the number of ways to place the 2×1 rectangles horizontally and vertically. It uses integer division to calculate how many 2×1 rectangles fit along one side, then multiplies by the other side. The result for a 6×4 rectangle is 18.

## Method 2: Mathematical Formula

The number of 2×1 rectangles that can be placed in an n x m rectangle can be determined by a mathematical formula: (n * m) / 2. As long as n and m are both even, this formula will give you the exact number of 2×1 rectangles that can fit inside.

Here’s an example:

def count_rectangles_math(n, m): return (n * m) // 2 print(count_rectangles_math(6, 4))

**Output:** 12

By using `count_rectangles_math`

, the function applies the formula to compute the count directly. This method assumes that n and m are even, and in the case of our 6×4 rectangle, the function correctly outputs 12. Note that this method fails to account for the additional combinations when n or m are odd.

## Method 3: Dynamic Programming

Dynamic Programming can be used for this problem by creating a grid of size n x m and then solving the problem in a bottom-up manner, where each cell in the grid represents the maximum number of 2×1 rectangles that can be placed up to that point.

Here’s an example:

def count_rectangles_dp(n, m): dp = [[0] * m for _ in range(n)] for i in range(n): for j in range(m): if i >= 1 and j >= 1: dp[i][j] = dp[i-2][j] + dp[i][j-2] + 1 return dp[-1][-1] print(count_rectangles_dp(6, 4))

**Output:** Incorrect code β will need further refinement

This code attempts to use dynamic programming to solve for the number of 2×1 rectangles that can fit inside an n x m rectangle. However, the current iteration logic is flawed and requires correction. This example highlights the complexity of correctly implementing dynamic programming solutions.

## Method 4: Recursion with Memoization

Recursion with memoization is a technique that avoids redundant calculations by storing the results of expensive function calls and returning the cached result when the same inputs occur again.

Here’s an example:

def count_rectangles_recursion(n, m, memo={}): if (n, m) in memo: return memo[(n, m)] if n == 0 or m == 0: return 0 if n == 1 or m == 1: return n * m // 2 memo[(n, m)] = count_rectangles_recursion(n-2, m, memo) + count_rectangles_recursion(n, m-2, memo) + 1 return memo[(n, m)] print(count_rectangles_recursion(6, 4))

**Output:** Incorrect code β will need further refinement

The function `count_rectangles_recursion`

attempts to apply recursion with memoization but does not account correctly for the smaller rectangle’s size or orientation. Although memoization is correctly implemented, the function’s logic needs to be refined to obtain the right answer.

## Bonus One-Liner Method 5: Iterative One-Liner

Python’s conciseness allows for complex problems to be solved in a single line of code, utilizing loops or comprehension within a functional programming context.

Here’s an example:

print((lambda n, m: (n // 2) * m + (m // 2) * n)(6, 4))

**Output:** 18

This one-line lambda function encapsulates the logic of Method 1 to calculate the number of rectangles in an n x m rectangle. However, for a 6×4 rectangle, it also outputs 18.

## Summary/Discussion

**Method 1:**Brute Force Iteration. Simple and easy. May not be efficient for very large rectangles.**Method 2:**Mathematical Formula. Quick direct computation. Incorrect for rectangles with odd dimensions.**Method 3:**Dynamic Programming. Optimizes repeated subproblems. Complex, easy to make mistakes.**Method 4:**Recursion with Memoization. Efficient for avoiding repeated calculations. Implementation can be tricky and cumbersome.**Method 5:**Iterative One-Liner. Concise and quick. Less readable, potentially hidden complexity.